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Title: Conformal blocks for Galois covers of algebraic curves
We study the spaces of twisted conformal blocks attached to a$\Gamma$-curve$\Sigma$with marked$\Gamma$-orbits and an action of$\Gamma$on a simple Lie algebra$\mathfrak {g}$, where$\Gamma$is a finite group. We prove that if$\Gamma$stabilizes a Borel subalgebra of$\mathfrak {g}$, then the propagation theorem and factorization theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed$\Gamma$-curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let$\mathscr {G}$be the parahoric Bruhat–Tits group scheme on the quotient curve$\Sigma /\Gamma$obtained via the$\Gamma$-invariance of Weil restriction associated to$\Sigma$and the simply connected simple algebraic group$G$with Lie algebra$\mathfrak {g}$. We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasi-parabolic$\mathscr {G}$-torsors on$\Sigma /\Gamma$when the level$c$is divisible by$|\Gamma |$(establishing a conjecture due to Pappas and Rapoport).
Burungale, Ashay A; Kobayashi, Shinichi; Ota, Kazuto(
, Journal of the Institute of Mathematics of Jussieu)
Abstract
LetKbe an imaginary quadratic field and$p\geq 5$a rational prime inert inK. For a$\mathbb {Q}$-curveEwith complex multiplication by$\mathcal {O}_K$and good reduction atp, K. Rubin introduced ap-adicL-function$\mathscr {L}_{E}$which interpolates special values ofL-functions ofEtwisted by anticyclotomic characters ofK. In this paper, we prove a formula which links certain values of$\mathscr {L}_{E}$outside its defining range of interpolation with rational points onE. Arithmetic consequences includep-converse to the Gross–Zagier and Kolyvagin theorem forE.
A key tool of the proof is the recent resolution of Rubin’s conjecture on the structure of local units in the anticyclotomic${\mathbb {Z}}_p$-extension$\Psi _\infty $of the unramified quadratic extension of${\mathbb {Q}}_p$. Along the way, we present a theory of local points over$\Psi _\infty $of the Lubin–Tate formal group of height$2$for the uniformizing parameter$-p$.
Khayutin, Ilya; Nelson, Paul D; Steiner, Raphael S(
, Forum of Mathematics, Pi)
Abstract
Letfbe an$L^2$-normalized holomorphic newform of weightkon$\Gamma _0(N) \backslash \mathbb {H}$withNsquarefree or, more generally, on any hyperbolic surface$\Gamma \backslash \mathbb {H}$attached to an Eichler order of squarefree level in an indefinite quaternion algebra over$\mathbb {Q}$. Denote byVthe hyperbolic volume of said surface. We prove the sup-norm estimate$$\begin{align*}\| \Im(\cdot)^{\frac{k}{2}} f \|_{\infty} \ll_{\varepsilon} (k V)^{\frac{1}{4}+\varepsilon} \end{align*}$$
with absolute implied constant. For a cuspidal Maaß newform$\varphi $of eigenvalue$\lambda $on such a surface, we prove that$$\begin{align*}\|\varphi \|_{\infty} \ll_{\lambda,\varepsilon} V^{\frac{1}{4}+\varepsilon}. \end{align*}$$
We establish analogous estimates in the setting of definite quaternion algebras.
Chan, William; Jackson, Stephen; Trang, Nam(
, Forum of Mathematics, Sigma)
Abstract
This paper will study almost everywhere behaviors of functions on partition spaces of cardinals possessing suitable partition properties. Almost everywhere continuity and monotonicity properties for functions on partition spaces will be established. These results will be applied to distinguish the cardinality of certain subsets of the power set of partition cardinals.
The following summarizes the main results proved under suitable partition hypotheses.
If$\kappa $is a cardinal,$\epsilon < \kappa $,${\mathrm {cof}}(\epsilon ) = \omega $,$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the almost everywhere short length continuity property: There is a club$C \subseteq \kappa $and a$\delta < \epsilon $so that for all$f,g \in [C]^\epsilon _*$, if$f \upharpoonright \delta = g \upharpoonright \delta $and$\sup (f) = \sup (g)$, then$\Phi (f) = \Phi (g)$.
If$\kappa $is a cardinal,$\epsilon $is countable,$\kappa \rightarrow _* (\kappa )^{\epsilon \cdot \epsilon }_2$holds and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the strong almost everywhere short length continuity property: There is a club$C \subseteq \kappa $and finitely many ordinals$\delta _0, ..., \delta _k \leq \epsilon $so that for all$f,g \in [C]^\epsilon _*$, if for all$0 \leq i \leq k$,$\sup (f \upharpoonright \delta _i) = \sup (g \upharpoonright \delta _i)$, then$\Phi (f) = \Phi (g)$.
If$\kappa $satisfies$\kappa \rightarrow _* (\kappa )^\kappa _2$,$\epsilon \leq \kappa $and$\Phi : [\kappa ]^\epsilon _* \rightarrow \mathrm {ON}$, then$\Phi $satisfies the almost everywhere monotonicity property: There is a club$C \subseteq \kappa $so that for all$f,g \in [C]^\epsilon _*$, if for all$\alpha < \epsilon $,$f(\alpha ) \leq g(\alpha )$, then$\Phi (f) \leq \Phi (g)$.
Suppose dependent choice ($\mathsf {DC}$),${\omega _1} \rightarrow _* ({\omega _1})^{\omega _1}_2$and the almost everywhere short length club uniformization principle for${\omega _1}$hold. Then every function$\Phi : [{\omega _1}]^{\omega _1}_* \rightarrow {\omega _1}$satisfies a finite continuity property with respect to closure points: Let$\mathfrak {C}_f$be the club of$\alpha < {\omega _1}$so that$\sup (f \upharpoonright \alpha ) = \alpha $. There is a club$C \subseteq {\omega _1}$and finitely many functions$\Upsilon _0, ..., \Upsilon _{n - 1} : [C]^{\omega _1}_* \rightarrow {\omega _1}$so that for all$f \in [C]^{\omega _1}_*$, for all$g \in [C]^{\omega _1}_*$, if$\mathfrak {C}_g = \mathfrak {C}_f$and for all$i < n$,$\sup (g \upharpoonright \Upsilon _i(f)) = \sup (f \upharpoonright \Upsilon _i(f))$, then$\Phi (g) = \Phi (f)$.
Suppose$\kappa $satisfies$\kappa \rightarrow _* (\kappa )^\epsilon _2$for all$\epsilon < \kappa $. For all$\chi < \kappa $,$[\kappa ]^{<\kappa }$does not inject into${}^\chi \mathrm {ON}$, the class of$\chi $-length sequences of ordinals, and therefore,$|[\kappa ]^\chi | < |[\kappa ]^{<\kappa }|$. As a consequence, under the axiom of determinacy$(\mathsf {AD})$, these two cardinality results hold when$\kappa $is one of the following weak or strong partition cardinals of determinacy:${\omega _1}$,$\omega _2$,$\boldsymbol {\delta }_n^1$(for all$1 \leq n < \omega $) and$\boldsymbol {\delta }^2_1$(assuming in addition$\mathsf {DC}_{\mathbb {R}}$).
Matomäki, Kaisa; Shao, Xuancheng; Tao, Terence; Teräväinen, Joni(
, Forum of Mathematics, Pi)
Abstract
We study higher uniformity properties of the Möbius function$\mu $, the von Mangoldt function$\Lambda $, and the divisor functions$d_k$on short intervals$(X,X+H]$with$X^{\theta +\varepsilon } \leq H \leq X^{1-\varepsilon }$for a fixed constant$0 \leq \theta < 1$and any$\varepsilon>0$.
More precisely, letting$\Lambda ^\sharp $and$d_k^\sharp $be suitable approximants of$\Lambda $and$d_k$and$\mu ^\sharp = 0$, we show for instance that, for any nilsequence$F(g(n)\Gamma )$, we have$$\begin{align*}\sum_{X < n \leq X+H} (f(n)-f^\sharp(n)) F(g(n) \Gamma) \ll H \log^{-A} X \end{align*}$$
As a consequence, we show that the short interval Gowers norms$\|f-f^\sharp \|_{U^s(X,X+H]}$are also asymptotically small for any fixedsfor these choices of$f,\theta $. As applications, we prove an asymptotic formula for the number of solutions to linear equations in primes in short intervals and show that multiple ergodic averages along primes in short intervals converge in$L^2$.
Our innovations include the use of multiparameter nilsequence equidistribution theorems to control type$II$sums and an elementary decomposition of the neighborhood of a hyperbola into arithmetic progressions to control type$I_2$sums.
Ergemlidze, Beka; Molla, Theodore(
, Combinatorics, Probability and Computing)
Abstract
For a subgraph$G$of the blow-up of a graph$F$, we let$\delta ^*(G)$be the smallest minimum degree over all of the bipartite subgraphs of$G$induced by pairs of parts that correspond to edges of$F$. Johansson proved that if$G$is a spanning subgraph of the blow-up of$C_3$with parts of size$n$and$\delta ^*(G) \ge \frac{2}{3}n + \sqrt{n}$, then$G$contains$n$vertex disjoint triangles, and presented the following conjecture of Häggkvist. If$G$is a spanning subgraph of the blow-up of$C_k$with parts of size$n$and$\delta ^*(G) \ge \left(1 + \frac 1k\right)\frac n2 + 1$, then$G$contains$n$vertex disjoint copies of$C_k$such that each$C_k$intersects each of the$k$parts exactly once. A similar conjecture was also made by Fischer and the case$k=3$was proved for large$n$by Magyar and Martin.
In this paper, we prove the conjecture of Häggkvist asymptotically. We also pose a conjecture which generalises this result by allowing the minimum degree conditions in each bipartite subgraph induced by pairs of parts of$G$to vary. We support this new conjecture by proving the triangle case. This result generalises Johannson’s result asymptotically.
@article{osti_10519991,
place = {Country unknown/Code not available},
title = {Conformal blocks for Galois covers of algebraic curves},
url = {https://par.nsf.gov/biblio/10519991},
DOI = {10.1112/S0010437X23007418},
abstractNote = {We study the spaces of twisted conformal blocks attached to a$\Gamma$-curve$\Sigma$with marked$\Gamma$-orbits and an action of$\Gamma$on a simple Lie algebra$\mathfrak {g}$, where$\Gamma$is a finite group. We prove that if$\Gamma$stabilizes a Borel subalgebra of$\mathfrak {g}$, then the propagation theorem and factorization theorem hold. We endow a flat projective connection on the sheaf of twisted conformal blocks attached to a smooth family of pointed$\Gamma$-curves; in particular, it is locally free. We also prove that the sheaf of twisted conformal blocks on the stable compactification of Hurwitz stack is locally free. Let$\mathscr {G}$be the parahoric Bruhat–Tits group scheme on the quotient curve$\Sigma /\Gamma$obtained via the$\Gamma$-invariance of Weil restriction associated to$\Sigma$and the simply connected simple algebraic group$G$with Lie algebra$\mathfrak {g}$. We prove that the space of twisted conformal blocks can be identified with the space of generalized theta functions on the moduli stack of quasi-parabolic$\mathscr {G}$-torsors on$\Sigma /\Gamma$when the level$c$is divisible by$|\Gamma |$(establishing a conjecture due to Pappas and Rapoport).},
journal = {Compositio Mathematica},
volume = {159},
number = {10},
publisher = {Cambridge University Press},
author = {Hong, Jiuzu and Kumar, Shrawan},
}
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